Sappington, Charles B.

English

Research Papers in Economics

SOUTHERN  JOURNAL  OF AGRICULTURAL  ECONOMICS  December,  1970
A NUMERICAL EXAMPLE OF THE PRACTICAL
USE OF DUMMY  VARIABLES
Charles  Sappington*
Although  the use of dummy variables in regression  ANALYSES
analysis is quite common, the implications  of alterna-
tive  models  for  incorporating  dummy  variables  are
not generally understood. References  dealing with the  Various  models  can  be  rationalized  from  these
use of dummy variables are numerous but scattered in  data;  some  presented  here  are  examples  of  proper
the literature.  The purpose of this article is to demon-  analysis  under  given  situations  and  others  are
strate, using numerical examples, the implications and  examples generally considered improper.
interrelationships  among various  models  which incor-
porate  dummy  variables.  Five  separate  models  are  Allmodelsareofthegeneralform  P =a+bQ.In
considered.  -- some,  dummy variables  are added  to  take account of
the  qualitative  variable.  The  several  models  con-
irDATA  ''-  sidered here  are  of little interest standing alone; how-
ever, much can be learned by comparison.
Hypothetical  data  were  generated  from  three
straight lines  representing  the  "true"  demand  sched-  DefintionofVariables
ules  for  each  of  three  different  sizes  of  potatoes:
small, medium, and large. The formulae  are:
The variables used in all models are defined as:
For Small:  P= 12.0 - .0100  (1)
P  =  price of potatoes (cents/lb.)
For Medium:  P= 22.0 - .0067Q  (2)
Q  =  quantity  of  potatoes  sold (lbs.)  in that par-
For Large:  P= 20.0 -. 0125Q  (3)  ticular size group
where  X2=  1 if in small group,  otherwise = 0
P  = price  X3= 1 if in medium group, otherwise  = 0
Q  =  quantity  X4=  1 if in large  group, otherwise  = 0
These  lines  have different  slopes but  do not intersect  X5=  1 if in small group
over  the  range  of  the  data.  These  three  equations
indicate  that the demand  for potatoes is a function of  2 if in medium group
price,  quantity,  and  the  qualitative  variable,  size.
Price  is considered  as the dependent variable  through-  3 if in large  group
out this paper.
X6=  X3Q;  i.e.,  =Q of the  medium  size if in  the
To  these  "true"  price  readings  a  small  random  medium size group,  otherwise  = 0
error,  drawn  from  a  rectangular  distribution,  was
added.  This  process  generated  the  combination  time  X7=  X4Q; i.e., = Q of the large  size if the large
series by size group data of Table  I.  size group, otherwise  = 0
*Charles Sappington  is assistant  professor of Agricultural  Economics, University of Tennessee.  Helpful suggestions were made by
L. L. Bauer,  L. H.  Keller,  J. G.  Snell, and B. J. Trevena.
197Model  I  For Small:  (4)
This  model involves the independent estimation of
an equation  for each size  group.  This is equivalent to  P = 13.098-.0135Q (R2 .7887;  u= .3199)
assuming  that  each  size  group  is  really  a  separate
product  with the  demand  for  each  a function  of its  For Medium:  (5)
price  and  quantity.  Such an  analysis  of time  series, 
qualitative  data  is proper,  the simplest  to make,  and  P= 21.066 - .0053Q (R2 = .1112;  = .6472)
may be of direct use  or suggest what further analysis
may be useful.  For Large:  (6)
A
The separate demand functions are:  P = 22.395-0157Q  (R2 = .7042;  = .7109)
TABLE  1.  TIME SERIES DATA  GENERATED  FOR THREE DIFFERENT SIZES OF POTATOES
Time  Potatoes
Periods  Small  Medium  Large
P(cents/lb)  Q(lbs)  P(cents/lb)  Q(lbs)  P(cents/lb)  Qtlbs)
1  9.50  300  18.18  600  11.00  800
2  9.15  325  18.75  575  8.05  900
3  9.40  250  17.61  625  8.06  875
4  8.70  300  18.64  650  9.31  775
5  8.20  350  16.91  700  10.29  825
6  9.65  275  18.51  550  11.94  725
7  8.50  300  18.78  600  11.65  700
8  9.80  250  18.38  600  9.40  800
9  8.45  325  18.01  625  8.45  900
10  8.35  325  17.51  550  9.46  875
11  10.00  250  17.58  600  8.57  850
12  9.10  300  17.04  650  9.91  775
13  7.80  400  17.71  550  11.02  750
14  8.15  375  16.98  600  9.20  800
15  9.10  300  17.18  600  10.65  700
Mean  8.923  308.33  17.842  605.00  9.797  803.33
T.S.S.  6.2944  6.1278  22.204
St. dev.  .6705  43.98  .6616  41.40  1.2594  67.39
All Groups
P(cents/lb)  Q(lbs)
Mean  12.188  572.222
T.S.S.  759.88
St. dev.  4.155  211.946
198By  comparing Equations (4), (5), and (6) with (1),  Model  1 only the variation  within each  size  group is
(2), and (3), respectively,  it  can be seen that the esti-  relevant  (Table  1).  The increased  number of degrees
mated  equations  of this model  closely  approximate  of freedom  for  Model  2  as  compared  with  Model  1
the  "true"  equations to which the random  error was  (41  vs  13)  is a  strong argument  in favor of Model  2.
added.  Further,  these  three  estimates  are  the  best  However,  the  imposed  constraint  did  result  in  pa-
possible  estimates  of the  true  parameters  obtainable  rameter  estimates quite  different  from  the  "best"
using ordinary least squares.  estimates of Model  1.1
Model 2  Model  3
Whereas,  Model  1 was three  separate regressions of  This  model  removes  the  constraint  of a  common
15  observations,  the  next  three  models  use  all  45  slope  imposed  on Model  2.  Variables X6 and X7 are
observations  in  one  regression  and  consider  various  added  so  as  to allow  for  separate  slopes  as  well  as
techniques  of  incorporating  dummy  variables  to  different  intercept values for each demand curve.  This
separate  the  qualitative aspects  of the data. In Model  analysis  is  statistically  equivalent  to  making  no
2, the particular  dummy variable  used is the usual (0,  assumption  about  the  slopes  or  the  intercepts.  The
1) type which  allows  for intercept changes only.  This  procedure  is  economically  equivalent  to  assuming
procedure  is equivalent  to assuming  that there  is one  that  there  is one product  with important  differences
product  with  important  differences  among  size  among size  groups which affect  not only the vertical
groups.  Using  this  procedure,  three  parallel  demand  placement  of the  three  demand  functions, but  their
curves  are  estimated,  one  for  each  size  group.  Since  slopes  as  well.  The  effects  of  the  qualitative  dif-
the results of Model 1 indicate that the best estimates  ferences  are,  thus, allowed  a larger  role  in the  deter-
of  the  slopes  of  the  three  demand  curves  are  not  mination of the demand for potatoes.
equal, the results of this model are constrained.
The estimating equation for this model is:
The  dummy variable  for  the'small group is used as  A
the  base; i.e.,  X2 is  deleted  to avoid  singularity.  The  P =  13.098  -. 0135Q  + 7.968X3 +  (11)
constant  term  is,  thus,  the  true  unknown  intercept
plushb2.  f  ;  9.297X4 +  .0082X6 - .00214X7
The estimating equation for this model is:  (R2 =.9824;  U  =  .2850)
P= 12.934-.0130Q  +12778X  +  (7)  The demand functions from (11)  are: P = 12:934 - .0130Q + 12.778X 3 +  (7)
For  Small:  (12)
7.313X4 (R2 = .9800;  = .6093)  ^
P= 13.098  - .0135Q
The  demand functions  from (7) are:
For Small:  (8)  For Medium:  (13)
A  ^
P= 12.934-.0130Q  P =  (13.098  + 7.968)+  (-.0135  + .0082)Q
For Medium:  (9)  =  21.066  - .0053Q
A
P= (12.934 + 12.778)-  .0130Q =or For Large:  (14)
25.712-  .0130Q  P =  (13.098  + 9.297)  + (-.0135  - .0021)Q
For Large:  (0  = 22.395  - .0156Q
P= (12.934 +7.313)-0130  =  Except  for  minor  rounding  errors,  these  param-
P* . . (123  eters  are  identical  to those  of Model  1. This method
20.247 - .0130Q  of analysis  repeats  the best estimates of Model  1 by
removing  all  constraints,  but  has  the  advantage of a
The  standard  error  is  approximately  equal to the  considerable  increase  in  R2 for  the  reasons  given
average of those of Model  1, but the  R2 is increased  under Model 2.
considerably.  This  is  to  be  expected  since  the  total
sum of squares  to be explainedin Model 2 is the sum  Comparing  the R2 and  standard  error, Model  3  is
of the variation within  and between groups,  while  in  slightly  superior  to Model 2.  The constraint of Model
1The  demand functions (8), (9), and (10) could be exactly duplicated using a (0, 9) dummy variable  rather than a (0,  1).
Had this been done, b3 and b4 would be  1/9 of their reported values; yet, nothing would really be changed.
1992 was effective since  the best estimate of the slopes of  For Medium:  (17)
the lines are  different.  Even though the constraint  of 
Model  2  is  only  slightly  effective,  the  sum  of the  P =  [9.824 - 4.338  (2)]  + .0193Q=
squared  residual  terms  of Model  3  (disregarding  the
two degrees  of freedom difference,  the standard error  1.148 + .0193Q
of the estimate)  is  necessarily  less than that of Model
2 since  a  constrained minimum can never be less than  For Large:  (18)
an unconstrained minimum.
P = [9.824 - 4.338  (3)]  +.0193Q=
The added value of a higher R2 and lower standard
error  is,  however,  not  without  cost.  While  not  re-  -3.190 +.0193Q
ported  here,  the  standard  errors  of  b3 and  b4 in
Model  3  are over  twice  those  of b3 and b4 in Model  Given  the  data  used  here,  these  results  are  com-
2.  This is caused by the high degree  of linear relation-  pletely unacceptable;  even the slopes change sign.  If a
ship between X3 and X6 on the one hand and X4 and  student  obtained  such  results  as  these,  the  impulse
X7 on the other (r3,6 and r4,7 both exceed  .99). The  would likely  be to find another problem.  Instead, he
impact  of multicollinearity  on the estimated standard  should  simply  remove  or  relax  the  constraint  as  in
errors  of the  coefficients  is  a  very  real drawback  of  Model 2 or 3.
this  type  model.  Johnston  [2,  pp.  205-206]  shows
that  an increasing  degree of multicollinearity  can also  Admittedly, we  rigged our  data so that this paral-
affect the  estimated  regression coefficients. However,  lel,  equidistant  and  ordered  constraint  would  be
such was not the case for Model 3.  severe  since the results of Model 2 indicate that, given
common  slopes,  the estimated  curve  for the  medium
class  lies  above  both  small  and  large.  We  did  not
1Model  4  -realize  ex  ante just  how  severe  the rigging was.  Per-
-•:~~~~~~  - - ~haps,  though,  the  message  is  made  clearer  this way.
Had  we  exchanged  our  arbitrarily  assigned  numbers
This model  is  presented-  in  an  effort to warn  the  for medium and large; i.e., a (1,3, 2) rather than a (1,
novice  of the danger of doing what seems to be a first  2,3) dummy variable, the constraint would have been
impulse when qualitative  data are to be analyzed. The  lessened considerably.  With the (1,3,2) dummy va-
impulse  seems  to  be  to assign  some  number to each  able,  the  size ordering would  be correct but the equi-
size group, often with invalid reasoning.  distant  constraint  would  be mildly more severe than
that of Model 2.
The impulse with these data is to assign  1, 2, and 3
to small,  medium,  and large,  respectively.  Using a (1,  The  demand  functions (16),  (17), and (18) could
2,  3)  dummy  variable  for  intercept  changes alone  is  be  exactly  duplicated  using  a  (7,  8,  9)  dummy vari-
statistically  equivalent  to  asserting  that  the  three  able  rather than  the (1,  2,3). Hadthisbeendone, b
demand  curves  are  not  only  parallel  but  also  equi-  would  take  on  the  same  value  as in this model  but
distant  apart,  with  the  curve  for  the medium group  value  of the  constant  term,  a, would the  computed  value  of the  constant  term,  a,  would placed between the other two.  Since  this is not true, change. the  constraint  is  much  more  severe  than  that  of
Model  2.  In  general,  the  equidistant  parallel  and  Model 5
ordered  constraint  are  quite  strong  and  should  be
avoided.  This model treats  the data  as though they were all
time  series.  This  is the  most  severe  constraint  of all
The estimating equation for this model is:  those  discussed.  The  procedure  is  equivalent  to
assuming  that there  is  one demand  curve which  is a
P = 9.824 + .0193Q - 4.338X5 (15)  function of price and quantity alone.
(R2 = .0756;  u  = 4.089)  The estimating equation for this model is:
P = 10.473 + .003Q  (R2 = .0234;  u  =  (19)
The demand functions from (15)  are:
4.1540)
For Small:  (16)
.A  Equation  (19)  would  be  the  demand  curve  for  all
P =  [9.824 - 4.338  (1)]  + .0193Q =  three size groups.
5.486 + .0193Q  These  results  are  poor  by  any  standards.  The
200demand  curve  is  positively  sloped  and  the  R2 is  factory results than with no constraint.  Ordinary least
almost nonexistent.  squares is, of course,  constrained  in that it is linear in
the parameters.  We speak  here of constraints in addi-
If  no  size  differences  are  assumed  when  in  fact  tion  to  the  usual  ones.  These  unsatisfactory  results
there  are  differences,  the  time  series  data  should  be  can  vary  from  mild  to  severe  dependent  on  how
aggregated  across  the size groups yielding  15  observa-  severe  the  "subject  to"  is  to  the  data.  A  model,
tions.  If data are  both time series and qualitative, the  proper  in  one instance,  may  be improper  in another.
proper analysis must take account of both aspects. To  The  choice  of a  proper  model  is  dependent  on  the
assume  one portion of the data  away can,  and likely  view taken as to how the data are generated.
will,  give  spurious  results  as  to the  parameters  and
lead  to false conclusions.  The  five  models presented use data which are time
series  and  qualitative  in  nature.  Had  the size  groups
Other Possible  Models  been called  stores  A.  B,  and  C or states X,  Y, and Z,
the  data  would  have  been  combination  time  series
There are other models which might be of interest,  and  cross sectional.  The  same  comments and conclu-
One is  to use a  common intercept term but allow for  sions would apply.
slope changes. The model would be:
A  These  conclusions are:
P=a+b1 Q+b 6 X6 +b7X7
(1)  Independent  estimation  of each qualitative  or
A  second  is  to  specify  a  zero  intercept  term  and  cross  sectional  group  will  give the  best possible  esti-
duplicate  the  results  of Model  2.  The  model  would  mates  of  the  parameters.  The  R2 will  probably  be
dutbe:  -lower  than with other proper methods and degrees of be:
freedom may be a problem.
A
P  b1Q +b2X2  + b3X3 + b  X P =bQ+b 2 2 + bX 3 +b4 4 (2)  The  (0,  1) dummy  variable  to allow for  inter-
cept  changes  is  proper  only  if  some  a  priori  know- This  computation  involves  a  cross  product  of raw This comput  n  i  e  a  cs  p  t  of  r:  ledge  exists to justify the assertion that the functions data  rather  than  deviations  from  the  mean,  so  no  the  are parallel. deletion is  necessary  to avoid singularity  (the dummy  arepara
variables  are orthogonal).  Here, the computed R2 will
be  greater than that of Model 2 since the total sum of  (3)  Using  (0,  1) dummy  variables  on both  slope
squarees  is  tPi2  rather  than  o(P-  P)2.  Model  3  sct  of  and intercept gives, with the data used here, estimates
scould be duplicated similarly.  Mdl  3  of the  parameters  as good as those  discussed  in  con-
clusion  number  (1).  However,  multicollinearity  will
make the testing of the bi values misleading.
CONCLUSIONS
(4)  Using a  (1,  2, 3) dummy variable should gener-
The  main consideration  of this paper is alternative  ally be avoided.
methods  of  handling  a  qualitative  variable.  When
numbers  are  assigned to such variables, some thought  (5)  When  data  are  time  series  as  well  as  either
must be  given to the imposition  of a constraint.  Any  qualitative  or cross  sectional in nature,  the  statistical
imposed  constraint,  if effective,  will  yield  less  satis-  procedure should take account of both aspects.
REFERENCES
1.  Ben-David,  Shaul  and  William  G.  Tomek,  Allowing for Slope and Intercept Changes in Regression Analysis,
AER 179, Cornell University,  Nov.  1965.
2.  Johnston, J., Econometric Methods, McGraw-Hill Book Company,  Inc.,  New York,  1963.
3.  Suits,  Daniel B., "Use of Dummy Variables in Regression Equation,"  J. of American Statistics Association, pp.
548-55 1, Dec.  1957.
4.  Tomek,  William  G.,  "Using Zero-One  Variables  with  Time  Series  Data in  Regression  Equations," J. of Farm
Econ., pp. 814-822, Nov.  1963
5.  University  of California,  BMD  Bromedical Computer Programs, BMDO2R program,  University  of California
Press, Los Angeles,  1968.
201

A NUMERICAL EXAMPLE OF THE PRACTICAL USE OF DUMMY VARIABLES

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